Transient times, resonances and drifts of attractors in dissipative rotational dynamics

In a dissipative system the time to reach an attractor is often influenced by the peculiarities of the model and in particular by the strength of the dissipation. In particular, as a dissipative model we consider the spin-orbit problem providing the dynamics of a triaxial satellite orbiting around a central planet and affected by tidal torques. The model is ruled by the oblateness parameter of the satellite, the orbital eccentricity, the dissipative parameter and the drift term. We devise a method which provides a reliable indication on the transient time which is needed to reach an attractor in the spin-orbit model; the method is based on an analytical result, precisely a suitable normal form construction. This method provides also information about the frequency of motion. A variant of such normal form used to parametrize invariant attractors provides a specific formula for the drift parameter, which in turn yields a constraint - which might be of interest in astronomical problems - between the oblateness of the satellite and its orbital eccentricity.Comment: 21 pages, 7 figures, colo

The orbit of 2010 TK7. Possible regions of stability for other Earth Trojan asteroids

Recently the first Earth Trojan has been observed (Mainzer et al., ApJ 731) and found to be on an interesting orbit close to the Lagrange point L4 (Connors et al., Nature 475). In the present study we therefore perform a detailed investigation on the stability of its orbit and moreover extend the study to give an idea of the probability to find additional Earth-Trojans. Our results are derived using different approaches: a) we derive an analytical mapping in the spatial elliptic restricted three-body problem to find the phase space structure of the dynamical problem. We explore the stability of the asteroid in the context of the phase space geometry, including the indirect influence of the additional planets of our Solar system. b) We use precise numerical methods to integrate the orbit forward and backward in time in different dynamical models. Based on a set of 400 clone orbits we derive the probability of capture and escape of the Earth Trojan asteroids 2010 TK7. c) To this end we perform an extensive numerical investigation of the stability region of the Earth's Lagrangian points. We present a detailed parameter study in the regime of possible stable tadpole and horseshoe orbits of additional Earth-Trojans, i.e. with respect to the semi-major axes and inclinations of thousands of fictitious Trojans. All three approaches underline that the Earth Trojan asteroid 2010 TK7 finds himself in an unstable region on the edge of a stable zone; additional Earth-Trojan asteroids may be found in this regime of stability.Comment: 11 pages, 16 figure

Kinematic models of the interplanetary magnetic field

Current knowledge on the description of the interplanetary magnetic field is reviewed with an emphasis on the kinematic approach as well as the analytic expression. Starting with the Parker spiral field approach, further effects are incorporated into this fundamental magnetic field model, including the latitudinal dependence, the poleward component, the solar cycle dependence, and the polarity and tilt angle of the solar magnetic axis. Further extensions are discussed in view of the magnetohydrodynamic treatment, the turbulence effect, the pickup ions, and the stellar wind models. The models of the interplanetary magnetic field serve as a useful tool for theoretical studies, in particular on the problems of plasma turbulence evolution, charged dust motions, and cosmic ray modulation in the heliosphere.</p

Evolution and stability of Laplace-like resonances under tidal dissipation

AbstractThe Laplace resonance is a configuration that involves the commensurability between the mean motions of three small bodies revolving around a massive central one. This resonance was first observed in the case of the three inner Galilean satellites, Io, Europa, and Ganymede. In this work the Laplace resonance is generalised by considering a system of three satellites orbiting a planet that are involved in mean motion resonances. These Laplace-like resonances are classified in three categories: first-order (2:1&2:1, 3:2&3:2, 2:1&3:2), second-order (3:1&3:1) and mixed-order resonances (2:1&3:1). In order to study the dynamics of the system we implement a model that includes the gravitational interaction with the central body, the mutual gravitational interactions of the satellites, the effects due to the oblateness of the central body and the secular interaction of a fourth satellite and a distant star. Along with these contributions we include the tidal interaction between the central body and the innermost satellite. We study the survival of the Laplace-like resonances and the evolution of the orbital elements of the satellites under the tidal effects. Moreover, we study the possibility of capture into resonance of the fourth satellite

Effective resonant stability of Mercury

Mercury is the unique known planet that is situated in a 3:2 spin-orbit resonance nowadays. Observations and models converge to the same conclusion: the planet is presently deeply trapped in the resonance and situated at the Cassini state 1, or very close to it. We investigate the complete non-linear stability of this equilibrium, with respect to several physical parameters, in the framework of Birkhoffnormal form and Nekhoroshev stability theory. We use the same approach we have adopted for the 1:1 spin-orbit case with a peculiar attention to the role of Mercury's non-negligible eccentricity. The selected parameters are the polar moment of inertia, the Mercury's inclination and eccentricity and the precession rates of the perihelion and node. Our study produces a bound to both the latitudinal and longitudinal librations (of 0.1 rad) for a long but finite time (greatly exceeding the age of the Solar system). This is the so-called effective stability time. Our conclusion is that Mercury, placed inside the 3:2 spin-orbit resonance, occupies a very stable position in the space of these physical parameters, but not the most stable possible one

Enhancement of CO2/CH4 separation performances of 6FDA-based co-polyimides mixed matrix membranes embedded with UiO-66 nanoparticles

Metal-organic frameworks (MOFs) incorporation into mixed matrix membranes (MMMs) is gaining more attention due to the combined advantages of high separation performance and easy processability. Nanoparticles (NPs) of CO2-philic MOF UiO-66 (Zr-BDC) were synthesized with high surface area and ca. 50 nm particle size (and also for comparison with 100 and 200 nm sizes). They were incorporated into three 6FDA-based co-polyimides (namely 6FDA-BisP, 6FDA-ODA, and 6FDA-DAM), forming MMMs with loadings in the 4–23 wt% range. The NPs and MMMs were characterized accordingly by XRD, BET, SEM, TEM, FTIR, and TGA. CO2 and CH4 isotherms on the NPs were measured by a static volumetric method at the pressure up to 10 bar. Fractional free volume (FFV) was calculated using solid density, measured by pycnometer. Gas separation performance was evaluated using a feed composition of 50%:50% CO2:CH4 binary mixture at 35 °C and a pressure difference of 2 bar. The presence of UiO-66 NPs in the continuous 6FDA-BisP and 6FDA-ODA co-polyimides improved both CO2 permeability and CO2/CH4 selectivity by 50–180% and 70–220%, respectively. In the case of 6FDA-DAM MMMs, the CO2 permeability was significantly improved by 92%, while maintaining the CO2/CH4 selectivity. The best results in terms of CO2/CH4 selectivity were 41.9 for 6FDA-BisP (17 wt% filler loading, 108 Barrer of CO2), 57.0 for 6FDA-ODA (7 wt% filler loading, 43.3 Barrer of CO2) and 32.0 for 6FDA-DAM (8 wt% filler loading, 1728 Barrer of CO2)

An Overview of the 13:8 Mean Motion Resonance between Venus and Earth

It is known since the seminal study of Laskar (1989) that the inner planetary system is chaotic with respect to its orbits and even escapes are not impossible, although in time scales of billions of years. The aim of this investigation is to locate the orbits of Venus and Earth in phase space, respectively to see how close their orbits are to chaotic motion which would lead to unstable orbits for the inner planets on much shorter time scales. Therefore we did numerical experiments in different dynamical models with different initial conditions -- on one hand the couple Venus-Earth was set close to different mean motion resonances (MMR), and on the other hand Venus' orbital eccentricity (or inclination) was set to values as large as e = 0.36 (i = 40deg). The couple Venus-Earth is almost exactly in the 13:8 mean motion resonance. The stronger acting 8:5 MMR inside, and the 5:3 MMR outside the 13:8 resonance are within a small shift in the Earth's semimajor axis (only 1.5 percent). Especially Mercury is strongly affected by relatively small changes in eccentricity and/or inclination of Venus in these resonances. Even escapes for the innermost planet are possible which may happen quite rapidly.Comment: 14 pages, 11 figures, submitted to CMD

Where are the Uranus Trojans?

The area of stable motion for fictitious Trojan asteroids around Uranus' equilateral equilibrium points is investigated with respect to the inclination of the asteroid's orbit to determine the size of the regions and their shape. For this task we used the results of extensive numerical integrations of orbits for a grid of initial conditions around the points L4 and L5, and analyzed the stability of the individual orbits. Our basic dynamical model was the Outer Solar System (Jupiter, Saturn, Uranus and Neptune). We integrated the equations of motion of fictitious Trojans in the vicinity of the stable equilibrium points for selected orbits up to the age of the Solar system of 5 billion years. One experiment has been undertaken for cuts through the Lagrange points for fixed values of the inclinations, while the semimajor axes were varied. The extension of the stable region with respect to the initial semimajor axis lies between 19.05 < a < 19.3 AU but depends on the initial inclination. In another run the inclination of the asteroids' orbit was varied in the range 0 < i < 60 and the semimajor axes were fixed. It turned out that only four 'windows' of stable orbits survive: these are the orbits for the initial inclinations 0 < i < 7, 9 < i < 13, 31 < i < 36 and 38 < i < 50. We postulate the existence of at least some Trojans around the Uranus Lagrange points for the stability window at small and also high inclinations.Comment: 15 pages, 12 figures, submitted to CMD

Chirikov Diffusion in the Asteroidal Three-Body Resonance (5,-2,-2)

The theory of diffusion in many-dimensional Hamiltonian system is applied to asteroidal dynamics. The general formulations developed by Chirikov is applied to the Nesvorn\'{y}-Morbidelli analytic model of three-body (three-orbit) mean-motion resonances (Jupiter-Saturn-asteroid system). In particular, we investigate the diffusion \emph{along} and \emph{across} the separatrices of the (5,-2,-2) resonance of the (490) Veritas asteroidal family and their relationship to diffusion in semi-major axis and eccentricity. The estimations of diffusion were obtained using the Melnikov integral, a Hadjidemetriou-type sympletic map and numerical integrations for times up to $10^{8}$ years.Comment: 27 pages, 6 figure